# Linear representation theory of groups of order 12

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 12.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 12

## The list

Group Second part of GAP ID (GAP ID is (12,second part)) Linear representation theory page Linear representation theory page (family contexts)
dicyclic group:Dic12 1 linear representation theory of dicyclic group:Dic12 linear representation theory of dicyclic groups
cyclic group:Z12 2 linear representation theory of cyclic group:Z12 linear representation theory of cyclic groups
alternating group:A4 3 linear representation theory of alternating group:A4 linear representation theory of alternating groups, linear representation theory of projective special linear group of degree two over a finite field
dihedral group:D12 4 linear representation theory of dihedral group:D12 linear representation theory of dihedral groups
direct product of Z6 and Z2 5 linear representation theory of direct product of Z6 and Z2 linear representation theory of finite abelian groups

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

### Full listing

Group Second part of GAP ID (GAP ID is (12,second part)) Degrees of irreducible representations as list Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Total number of irreducible representations = number of conjugacy classes
dicyclic group:Dic12 1 1,1,1,1,2,2 4 2 0 6
cyclic group:Z12 2 1,1,1,1,1,1,1,1,1,1,1,1 12 0 0 12
alternating group:A4 3 1,1,1,3 3 0 1 4
dihedral group:D12 4 1,1,1,1,2,2 4 2 0 6
direct product of Z6 and Z2 5 1,1,1,1,1,1,1,1,1,1,1,1 12 0 0 12

### Grouping by degrees of irreducible representations

Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3 Total number of irreps = number of conjugacy classes Number of groups with these degrees of irreps Description of groups List of groups List of GAP IDs (second part)
12 0 0 12 2 the abelian groups of order 12 cyclic group:Z12, direct product of Z6 and Z2 2,5
4 2 0 6 2 the dihedral and dicyclic group dicyclic group:Dic12, dihedral group:D12 1,4
3 0 1 4 1 the alternating group alternating group:A4 3